How do I stop both my path animations starting at the same time?
The option Fixed Position let you control the position of the car from the starting point of the path (0) to the end (1).

To animate the position of the truck, go to frame 300 and set the value to 0, then press I to add a keyframe.
Do the same for the last frame of the animation and set the value to 1.

This method allows you to use the same path as a "Follow path" constraint for multiple objects.